Improved Polynomial Fuzzy Modeling and Controller with Stability Analysis for Nonlinear Dynamical Systems

Table 1

Slack expressions defined on subregions and obtained polynomial membership functions for Example 1.

Subregion; (𝑆𝑘)	Slack expression 𝐿𝑆𝑘(𝐱)	Membership function; 𝑝1(𝑥1)	Membership function; 𝑝2(𝑥1)
𝑆1: −3<𝑥1<−1	(𝑥1+3)(𝑥1+1)	0.2454−0.2081(𝑥1+2)+0.2273(𝑥1+2)^2	0.7546+0.2081(𝑥1+2)−0.2273(𝑥1+2)^2
𝑆2: −2<𝑥1<0	(𝑥1+2)𝑥1	0.1793+0.2702(𝑥1+1)+0.2104(𝑥1+1)^2	0.8207−0.2702(𝑥1+1)−0.2104(𝑥1+1)^2
𝑆3: −1<𝑥1<1	(𝑥1+1)(𝑥1−1)	0.4895+0.2003𝑥1−0.0833(𝑥1)^2	0.5105−0.2003𝑥1+0.0833(𝑥1)^2
𝑆4: 0<𝑥1<2	𝑥1(𝑥1−2)	0.8201+0.2708(𝑥1−1)−0.211(𝑥1−1)^2	0.1799−0.2708(𝑥1−1)+0.211(𝑥1−1)^2
𝑆5: 1<𝑥1<3	(𝑥1−1)(𝑥1−3)	0.7539−0.2085(𝑥1−2)−0.2277(𝑥1−2)^2	0.2461+0.2085(𝑥1−2)+0.2277(𝑥1−2)^2